Foundations — Spontaneity — second law; entropy ΔS

This page assumes nothing. If the parent note wrote a symbol, we build it here from the ground up, in an order where each idea leans only on the ones before it. If you can read a thermometer and know that $3\times 4=12$, you can start at line one.

0. The alphabet of the topic (map first)

Before the words, here is how the pieces stack. Read bottom-up: things at the top of an arrow are needed before the thing they point to. Notice every box below is written in plain words — the exact symbols ($k_B$, $q_{rev}$, $\Delta H$) get earned in the numbered sections that follow, so nothing appears here before its definition.

Now we define each box, in order.

1. A "state" and a "microstate" — the thing we count

Picture four coins on a table. The macrostate "2 heads showing" is one description. But it can be realised by several microstates: HHTT, HTHT, HTTH, THHT, THTH, TTHH — six different exact arrangements that all look like "2 heads".

Read the figure below: the horizontal axis lists each macrostate (0, 1, 2, 3, 4 heads); the bar height is how many microstates build it. The tallest bar sits at "2 heads" (height 6, coral) — the messy middle — while the tidy extremes "0 heads" and "4 heads" (lavender) have height 1 each. The coral arrow points to the peak to make the message loud: the most-arrangements macrostate is the one you'll almost always land in.

2. $W$ — the number of microstates

• Ordered, tidy states → small $W$ (only a few ways).
• Spread-out, messy states → huge $W$ (astronomically many ways).

The picture to hold: $W$ is how big the raffle is for a given outward state.

3. The logarithm $\ln$ — turning "multiply" into "add"

Before we can write $S=k_B\ln W$, we must earn the symbol $\ln$.

You do not need to love $e$ yet. The only property we use is this magic trick:

Read the figure below: the lavender curve is $y=\ln x$. Coral dots sit at $x=3$, $x=4$, and $x=12$. Follow the dashed guide-lines to the vertical axis: the height at $x=12$ (coral) is exactly the height at $x=3$ (mint) plus the height at $x=4$ (butter) stacked on top. That stacking-of-heights is the equation $\ln (3\times 4)=\ln 3+\ln 4$ drawn as a picture.

4. $S$ and $k_B$ — entropy, at last

5. $T$ — temperature, the "shaking strength"

Picture: $T$ is how violently the particles jiggle — the strength of nature's constant "shaking". Higher $T$ = harder shaking = more microstates get explored. We must use kelvin (never Celsius) in every entropy formula, because those formulas divide by $T$, and dividing by a possibly-zero or negative Celsius number would be nonsense.

6. $U$ — internal energy, the total jiggle-and-bond store

Before heat can be defined cleanly, we need the tank that heat pours into and out of.

The picture: $U$ is the water level in a tank. Two taps can change that level — heat ($q$) flowing in/out, and work ($w$) done on/by the system: $\Delta U=q+w\text{(First Law: energy is only moved, never created).}$

Why the topic needs $U$: heat and enthalpy are both defined relative to this energy tank, and the connection between microscopic $W$ and thermodynamic entropy (§7) runs through how $W$ grows when you pour energy $U$ in.

7. $q$ — heat, and why it's slippery

Read the figure below: both a mint straight line and a coral wiggly line connect the same START and END dots. The vertical "altitude" gained is identical for both — that stands for a state function ($U$, or $S$). But the heat exchanged along the way differs between the two routes — that is what "path function" means. The lavender arrow flags this: same endpoints, different $q$.

8. Reversible path, $q_{rev}$, and the bridge to $S=k_B\ln W$

$q_{rev}$ is the heat exchanged along that idealised gentlest path. It is the maximum heat you could draw for a given change, and — crucially — it makes $q_{rev}/T$ come out path-independent, i.e. a genuine state-function change.

The picture: $dS$ is a little slice of entropy change; $\Delta S$ (the big Greek delta = "change in") is all the slices added up along the reversible path.

9. $\Delta H$ — enthalpy, the heat you can weigh at constant pressure

We now build enthalpy from the energy tank $U$ (§6) rather than assert it.

• $\Delta H<0$: exothermic — heat released to surroundings.
• $\Delta H>0$: endothermic — heat absorbed from surroundings.

Because a beaker reaction runs at constant (atmospheric) pressure, its heat dump into the surroundings is $\Delta H$. That lets us find the surroundings' entropy change without measuring them directly: $\Delta S_{surr}=\frac{q_{surr}}{T}=\frac{-q_{sys}}{T}=\frac{-\Delta H_{sys}}{T}.$ (The minus sign: heat leaving the system enters the surroundings, so they receive $-\Delta H_{sys}$.) This links to First Law of Thermodynamics and later to Gibbs Free Energy ΔG.

10. The symbols $\ge$, $>$, $=$, $<$ in the Second Law

Read the figure below: it is a number line centred on $0$. The mint-shaded region to the right ($\Delta S_{univ}>0$) is labelled SPONTANEOUS with a green "go" arrow; the coral-shaded region to the left ($\Delta S_{univ}<0$) is labelled IMPOSSIBLE; the single point at $0$ (lavender) is EQUILIBRIUM. The Second Law is just this verdict: real processes must land on or to the right of zero.

11. Putting the alphabet together

Every symbol on the parent page now has a meaning, a picture, and a reason:

Prerequisites you may want to open next: Boltzmann distribution & microstates (where $W$ really comes from) and Third Law of Thermodynamics (what happens to $S$ as $T\to 0$). Return to the parent: the main topic note.

Equipment checklist